Problem: The arithmetic sequence $(a_i)$ is defined by the formula: $a_1 = 1$ $a_i = a_{i-1} + 6$ What is $a_{19}$, the nineteenth term in the sequence?
Solution: From the given formula, we can see that the first term of the sequence is $1$ and the common difference is $6$ To find the nineteenth term, we can rewrite the given recurrence as an explicit formula. The general form for an arithmetic sequence is $a_i = a_1 + d(i - 1)$ . In this case, we have $a_i = 1 + 6(i - 1)$ To find $a_{19}$ , we can simply substitute $i = 19$ into the our formula. Therefore, the nineteenth term is equal to $a_{19} = 1 + 6 (19 - 1) = 109$.